The 60 rotational symmetries of the icosahedron form a group isomorphic to \(A_5\) . The full symmetry group of the icosahedron \(A_5\times \mathbb {Z} _2 \)  has order 120, obtained by introducing the reflection through the center of the icosahedron. This group is one of finitely many exceptional complex reflection groups.
For a \((n+1)\) -dimensional complex vector space \(V\) , a complex reflection group \(G\subseteq \textrm {GL}(V)\)  is a finite group that is generated by pseudoreflections: those elements that fix a hyperplane pointwise. These groups were studied and classified in [1]}. Considering the collection of hyperplanes, given by pseudoreflections of \(G\) , and projectivizing \(V\)  yields a hyperplane configuration in \(\mathbb {P}^n\) . In addition, the natural action of \(G\)  on \(\mathbb {P}^n\)  permutes the hyperplanes. Configurations admitted by complex reflection groups exhibit interesting characteristics [2]}. For \(n=2\) , we obtain a line configuration in \(\mathbb {P}^2\) . Line configurations have been studied extensively [3]}, [4]} yet continue to be subject of recent research [5]}, [6]}.
